Non-Elementary Integration · Superchallenge problem

Let $I_n=\displaystyle\int_0^{\pi/4}\tan^nx\,dx$. Since $\tan^nx=\tan^{n-2}x(\sec^2x-1)$, these satisfy $I_n+I_{n-2}=\dfrac1{n-1}$ for $n\ge2$. Given $I_1=\dfrac12\ln2$, find $I_5$.